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34.10 (a) 3(ln x)2 1 2 5 4 1 ; (b) ; (c) ; (e) x(1 + 2 ln x); ( f ) ; (g) ; (h) 2 cot x. ; (d) 4x 1 x x ln x (x 1)(x + 1) 5x 2 2x ln x
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34.11 (a)
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1 1 1 2 ln |x| + C; (b) ln |7x 2| + C; (c) ln |x 4 1| + C; (d) (ln x + 1)3/2 + C; (e) ln |ln x| + C; 3 7 4 3 1 1 3 + 2 ln |x| + 3 + C; (h) ln |tan x| + C; (i) 2 ln |1 x| + C = ln + C; ( f ) ln |1 sin 2x| + C; (g) x 2 2 2x (1 x)2 1 ( j) (ln x)2 + C. 2 4x 2 (3 x 2 ) 4 x2 ; (b) y = (x 2)4 3 x + 5 x2 + 4 4 x 1 ; 2 + x 2 3 x+5 x +4
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34.12 (a) y = (c) y =
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x 2 1 sin x (2x + 3)4
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34.14 (a) 12 ln 2; (b) ln 2 2 ln 3. 34.15 (a) ln 2 + ln 5; (b) ln 2; (c) ln 5; (d) 2 ln 5; (e) 7 . 24 1 1 ln 2; ( f ) ln 5; (g) (2 ln 2 + ln 5); (h) 7 ln 2. 2 3 34.19 2 ln 2. 34.20 See Fig. A-26.
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34.16 y = x 1.
34.17 ln 2
2 ln 2 . 3
Fig. A-26
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ANSWERS TO SUPPLEMENTARY PROBLEMS
34.21 (a) v = 34.22 (a) y =
67 1 2 t t + 6 ln |t| + 2; (b) + 12 ln 3. 2 2 y 2x + 1 x 1 ; (b) y = . ; (c) y x y 1 y(x 2 + y2 1) y 3xy + 3y3 2 34.23 1 . 3
1 34.25 (a) ln 3 ln 2; (b) ln 7. 4 34.26 (a) 0.6937714032; (b) 0.6928353604; (c) 0.6931502307. 34.27 (a) 0.5671432904; (b) 1.763222834.
35.11 (a) 1 1 ; (b) x; (c) x 4 ; (d) x 1+ln 3 ; (e) x 1; ( f ) x ln x; (g) 2x; (h) . x 3
e1/x ex (x 1) 1 + ln x ; (g) x 1 ; (h) (ln ) x ; (i) 2; 35.12 (a) e x ; (b) 2 ; (c) ( sin x)ecos x ; (d) ex sec2 ex ; (e) ; ( f ) ex x x x2 ( j) ex + e x . 35.13 (a) 1 3x 2 1 2x x +1 e + C; (b) e x + C; (c) ( ex 2)3 + C; (d) ecos x + C; (e) 3 + C; ( f ) 2ex/2 + C; (g) + C; 3 3 2 ln 3 +1 3 1 1 2x 4 (h) e3x + C; (i) ex ln (ex + 1) + C; ( j) + C; (k) e x + C. 3 3 ln 2 4 2x 2x 2x + yexy 1 2y2 ; (b) 1 + y x = 1 + y x ; (d) ; (e) cot x. ; (c) 2 y x(ey 1) 2y + xexy e sec2 ey x e (1 + x 4 ) y e e1/y ln 2 x 0.5ex 1 e 2 ; (c) (2 ln x)x (ln x) 1 ; (d) [1 + ln (ln x)](ln x)ln x ; 2 x 2x + 3 = . 2 (x + 1)(x + 2)
35.14 (a)
35.15 (a) (ln 3)(cos x)3sin x ; (b) (e) y 2 1 1 + x+1 x+2
35.16 (a)
1 ln 2 ; (b) ; (c) ln 2; (d) ee ; (e) 2. 3 3 e 2 (e 1). 2 35.18 (a) 2; (b) (e2 + 1). 35.19 (e 1).
35.17 (a) e 1; (b)
35.20 max = e at x =
1 , min = at x = . 2 e 2
35.21 nn enx .
35.22 (a) y = (2 cos x)esin x , y = 2esin x (cos2 x sin x); (b) 2 rad/sec. 35.23 (a) v = 3e3t + 1; (b) 2 + 35.24 y = 2x + 2. ln 3 ; (c) x = e3t + t 1. 3
35.25 See Fig. A-27.
35.26 In Fig. 35-1(a) and Fig. A-27(d), multiply the horizontal scale by 1/ln 2. 35.29 max = e 1 (at x = e), min = 0 (at x = 1). 35.30 (c) 2.718145927. (The correct approximation to 10 signi cant gures is 2.718281828.)
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